Advertisements
Advertisements
प्रश्न
If \[x - \frac{1}{x} = 3,\] find the values of \[x^2 + \frac{1}{x^2}\] and \[x^4 + \frac{1}{x^4} .\]
Advertisements
उत्तर
Let us consider the following equation: \[x - \frac{1}{x} = 3\]
Squaring both sides, we get:
\[\left( x - \frac{1}{x} \right)^2 = \left( 3 \right)^2 = 9\]
\[ \Rightarrow \left( x - \frac{1}{x} \right)^2 = 9\]
\[ \Rightarrow x^2 - 2 \times x \times \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 9\]
\[ \Rightarrow x^2 - 2 + \frac{1}{x^2} = 9\]
\[\Rightarrow x^2 + \frac{1}{x^2} = 11\] (Adding 2 to both sides)
Squaring both sides again, we get:
\[\left( x^2 + \frac{1}{x^2} \right)^2 = \left( 11 \right)^2 = 121\]
\[ \Rightarrow \left( x^2 + \frac{1}{x^2} \right)^2 = 121\]
\[ \Rightarrow \left( x^2 \right)^2 + 2\left( x^2 \right)\left( \frac{1}{x^2} \right) + \left( \frac{1}{x^2} \right)^2 = 121\]
\[ \Rightarrow x^4 + 2 + \frac{1}{x^4} = 121\]
\[\Rightarrow x^4 + \frac{1}{x^4} = 119\]
संबंधित प्रश्न
Subtract: 6xy from − 12xy
Subtract: a (b - 5) from b (5 - a)
Add the following algebraic expression:
3a2b, − 4a2b, 9a2b
Add the following algebraic expression:
\[\frac{2}{3}a, \frac{3}{5}a, - \frac{6}{5}a\]
Subtract:
\[\frac{2}{3} y^3 - \frac{2}{7} y^2 - 5 \text { from }\frac{1}{3} y^3 + \frac{5}{7} y^2 + y - 2\]
Subtract:
\[x^2 y - \frac{4}{5}x y^2 + \frac{4}{3}xy \text { from } \frac{2}{3} x^2 y + \frac{3}{2}x y^2 - \frac{1}{3}xy\]
Find the sum of the following expressions
7p + 6q, 5p – q, q + 16p
Add:
9ax + 3by – cz, –5by + ax + 3cz
Add the following expressions:
p2 – q + r, q2 – r + p and r2 – p + q
Write two different algebraic expressions for the word phrase “`(1/4)` of the sum of x and 7.”
